My recently deceased spouse's IRA has been rolled over into mine. Does the cost basis change for me?

The basis attributed to IRA assets that you inherited remains the same. Since you are a spouse beneficiary and you elect to treat the assets as your own by transferring them to your own IRA, the basis is now treated as if you made the non-deductible contribution. Going forward, you should file Form 8606 to determine the taxable portion of distributions you receive from any of your Traditional IRAs. If you also made non-deductible contributions to your IRA, the basis from the inherited assets should be combined with your basis.